Symmetry Resolved Entanglement Research Articles

Symmetry-resolved measures in quantum field theory: A short review

In this short review, we present the key definitions, ideas and techniques involved in the study of symmetry resolved entanglement measures, with a focus on the symmetry resolved entanglement entropy. In order to be able to define such entanglement measures, it is essential that the theory under study possess an internal symmetry. Then, symmetry-resolved entanglement measures quantify the contribution to a particular entanglement measure that can be associated to a chosen symmetry sector. Our review focuses on conformal (gapless/massless/critical) and integrable (gapped/massive) quantum field theories, where the leading computational technique employs symmetry fields known as (composite) branch point twist fields.

Disentangling the physics of the attractive Hubbard model as a fully interacting model of fermions via the accessible and symmetry-resolved entanglement entropies

Disentangling the physics of the attractive Hubbard model as a fully interacting model of fermions via the accessible and symmetry-resolved entanglement entropies

Entanglement of edge modes in (very) strongly correlated topological insulators

Identifying topological phases for a strongly correlated theory remains a non-trivial task, as defining order parameters, such as Berry phases, is not straightforward. Quantum information theory is capable of identifying topological phases for a theory that exhibits quantum phase transition with a suitable definition of order parameters that are related to different entanglement measures for the system. In this work, we study entanglement entropy for a coupled SSH model, both in the presence and absence of Hubbard interaction and at varying interaction strengths. For the free theory, edge entanglement acts as an order parameter, which is supported by analytic calculations and numerical (DMRG) studies. We calculate the symmetry-resolved entanglement and demonstrate the equipartition of entanglement for this model which itself acts as an order parameter when calculated for the edge modes. As the DMRG calculation allows one to go beyond the free theory, we study the entanglement structure of the edge modes in the presence of on-site Hubbard interaction for the same model. A sudden reduction of edge entanglement is obtained as interaction is switched on. The explanation for this lies in the change in the size of the degenerate subspaces in the presence and absence of interaction. We also study the signature of entanglement when the interaction strength becomes extremely strong and demonstrate that the edge entanglement remains protected. In this limit, the energy eigenstates essentially become a tensor product state, implying zero entanglement. However, a remnant entropy survives in the non-trivial topological phase, which is exactly due to the entanglement of the edge modes.

Refined symmetry-resolved Page curve and charged black holes* *Supported in part by the Natural Science Foundation of China (12035016, 12275275). It is also supported by the Beijing Natural Science Foundation (1222031) and the Innovative Projects of Science and Technology ( E2545BU210) at IHEP.

The Page curve plotted using the typical random state approximation is not applicable to a system with conserved quantities, such as the evaporation process of a charged black hole, during which the electric charge does not macroscopically radiate out with a uniform rate. In this context, the symmetry-resolved entanglement entropy may play a significant role in describing the entanglement structure of such a system. We attempt to impose constraints on microscopic quantum states to match the macroscopic phenomenon of charge radiation during black hole evaporation. Specifically, we consider a simple qubit system with conserved spin/charge serving as a toy model for the evaporation of charged black holes. We propose refined rules for selecting a random state with conserved quantities to simulate the distribution of charges during the different stages of evaporation and obtain refined Page curves that exhibit distinct features in contrast to the original Page curve. We find that the refined Page curve may have a different Page time and exhibit asymmetric behavior on both sides of the Page time. Such refined Page curves may provide a more realistic description for the entanglement between the charged black hole and radiation during the evaporation process.

Symmetry-resolved entanglement entropy, spectra & boundary conformal field theory

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

Symmetry-resolved entanglement: general considerations, calculation from correlation functions, and bounds for symmetry-protected topological phases

We discuss some general properties of the symmetry-resolved entanglement entropy in systems with particle number conservation. Using these general results, we describe how to obtain bounds on the entanglement components from correlation functions in Gaussian systems. We introduce majorization as an important tool to derive entanglement bounds. As an application, we derive lower bounds both for the number and the configurational entropy for chiral and Cn -symmetric topological phases. In some cases, our considerations also lead to an improvement of the previously known lower bounds for the entanglement entropy in such systems.

Symmetry-resolved entanglement in fermionic systems with dissipation

We investigate symmetry-resolved entanglement in out-of-equilibrium fermionic systems subject to gain and loss dissipation, which preserves the block-diagonal structure of the reduced density matrix. We derive a hydrodynamic description of the dynamics of several entanglement-related quantities, such as the symmetry-resolved von Neumann entropy and the charge-imbalance-resolved fermionic negativity. We show that all these quantities admit a hydrodynamic description in terms of entangled quasiparticles. While the entropy is dominated by dissipative processes, the resolved negativity is sensitive to the presence of entangled quasiparticles, and it shows the typical ‘rise and fall’ dynamics. Our results hold in the weak-dissipative hydrodynamic limit of large intervals, long times and weak dissipation rates.

Entanglement Resolution with Respect to Conformal Symmetry.

Entanglement is resolved in conformal field theory (CFT) with respect to conformal families to all orders in the UV cutoff. To leading order, symmetry-resolved entanglement is connected to the quantum dimension of a conformal family, while to all orders it depends on null vectors. Criteria for equipartition between sectors are provided in both cases. This analysis exhausts all unitary conformal families. Furthermore, topological entanglement entropy is shown to symmetry-resolve the Affleck-Ludwig boundary entropy. Configuration and fluctuation entropy are analyzed on grounds of conformal symmetry.

Nonequilibrium Full Counting Statistics and Symmetry-Resolved Entanglement from Space-Time Duality.

Owing to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution-or its Fourier transform known as full counting statistics (FCS)-contains much more information than say the mean value of the measured observable, and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables-the charged moments-that characterize how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a nontrivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out of equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain.

On the boundary conformal field theory approach to symmetry-resolved entanglement

We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved Rényi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with U(1)U(1), \mathbb{R}ℝ and \mathbb{Z}_2ℤ2 symmetry.

Full counting statistics and symmetry resolved entanglement for free conformal theories with interface defects

We consider the ground state of two species of one-dimensional critical free theories coupled together via a conformal interface. They have an internal U(1) global symmetry and we investigate the quantum fluctuations of the total charge on one side of the interface, giving analytical predictions for the full counting statistics, the charged moments of the reduced density matrix and the symmetry resolved Rényi entropies. Our approach is based on the relation between the geometry with the defect and the homogeneous one, and it provides a way to characterize the spectral properties of the correlation functions restricted to one of the two species. Our analytical predictions are tested numerically, finding a perfect agreement.

Symmetry resolved entanglement entropy in hyperbolic de Sitter space

In this paper, we study the relation between entanglement and global internal symmetries on de Sitter space. We consider two symmetric causally disconnected regions in the hyperbolic chart on de Sitter space. Since entanglement measures characterises correlations, the study of entanglement between the two causally disconnected regions gives information about the long range correlations in de Sitter space. When a theory possesses an additive global internal symmetry, the entanglement measures for a state with fixed global charge may be decomposed into local charge sectors in either subsystem and thus providing a finer resolution of entanglement. Here we will consider two theories: free complex scalar field, and free Dirac field on de Sitter space. Both theories possess global internal $U(1)$ symmetry. We study the symmetry resolved entanglement entropy for both theories in the Bunch-Davies vacuum state. We find that the symmetry resolved entanglement entropy has equipartition into local charge sectors upto the terms that scale as $V_{H_3}^0$ in the limit of large $V_{H_3}$, where $V_{H_3}$ is the volume of either region. This equipartition however is only broken by the terms of order $O(1/V_{H_3})$. Consequently, we have equipartition of symmetry resolved entanglement entropy in the limit of infinite volume.

Symmetry-resolved entanglement in critical non-Hermitian systems

The study of entanglement in the symmetry sectors of a theory has recently attracted a lot of attention since it provides better understanding of some aspects of quantum many-body systems. In this paper, we extend this analysis to the case of non-Hermitian models, in which the reduced density matrix $\rho_A$ may be non-positive definite and the entanglement entropy negative or even complex. Here we examine in detail the symmetry-resolved entanglement in the ground state of the non-Hermitian Su-Schrieffer-Heeger chain at the critical point, a model that preserves particle number and whose scaling limit is a $bc$-ghost non-unitary CFT. By combining bosonization techniques in the field theory and exact lattice numerical calculations, we analytically derive the charged moments of $\rho_A$ and $|\rho_A|$. From them, we can understand the origin of the non-positiveness of $\rho_A$ and naturally define a positive-definite reduced density matrix in each charge sector, which gives a well-defined symmetry-resolved entanglement entropy. As byproduct, we also obtain the analytical distribution of the critical entanglement spectrum.

Universal thermal corrections to symmetry-resolved entanglement entropy and full counting statistics

In the presence of a global symmetry, the entanglement entropy of a quantum system can be decomposed among the individual symmetry sectors, dubbed symmetry-resolved entanglement entropy. For a conformal field theory with Abelian symmetry, it obeys the equipartition theorem in the scaling limit, i.e., at the leading order, the entanglement is distributed equally between different symmetry sectors. In this work, we examine the thermal corrections to a single interval symmetry-resolved Rényi and entanglement entropies for two-dimensional conformal field theories on a circle. Using a low-temperature expansion of the thermal density matrix, we find that in addition to the mass gap and the degeneracy of the first excited state, these universal corrections depend also on the four-point correlation function of the primary fields. We also obtain thermal corrections to the full counting statistics of the ground state and define the probability fluctuations function which scales as {e}^{-2pi {Delta}_{psi}beta /L} , where ∆ψ is the scaling dimension of the lowest weight states. As an example, we explicitly evaluate the thermal corrections to the symmetry-resolved entanglement entropy and FCS for the spinless fermions and find a term breaking entanglement equipartition at order (log l)−2.

A path integral ground state Monte Carlo algorithm for entanglement of lattice bosons

A ground state path integral quantum Monte Carlo algorithm is introduced that allows for the study of entanglement in lattice bosons at zero temperature. The Rényi entanglement entropy between spatial subregions is explored across the phase diagram of the one dimensional Bose-Hubbard model for systems consisting of up to L=256L=256 sites at unit-filling without any restrictions on site occupancy, far beyond the reach of exact diagonalization. The favorable scaling of the algorithm is demonstrated through a further measurement of the Rényi entanglement entropy at the two dimensional superfluid-insulator critical point for large system sizes, confirming the existence of the expected entanglement boundary law in the ground state. The Rényi estimator is extended to measure the symmetry resolved entanglement that is operationally accessible as a resource for experimentally relevant lattice gases with fixed total particle number.

Entanglement resolution of free Dirac fermions on a torus

Whenever a system possesses a conserved charge, the density matrix splits into eigenspaces associated to the each symmetry sector and we can access the entanglement entropy in a given subspace, known as symmetry resolved entanglement (SRE). Here, we first evaluate the SRE for massless Dirac fermions in a system at finite temperature and size, i.e. on a torus. Then we add a massive term to the Dirac action and we treat it as a perturbation of the massless theory. The charge-dependent entropies turn out to be equally distributed among all the symmetry sectors at leading order. However, we find subleading corrections which depend both on the mass and on the boundary conditions along the torus. We also study the resolution of the fermionic negativity in terms of the charge imbalance between two subsystems. We show that also for this quantity, the presence of the mass alters the equipartition among the different imbalance sectors at subleading order.

Two-point functions of composite twist fields in the Ising field theory

All standard measures of bipartite entanglement in one-dimensional quantum field theories can be expressed in terms of correlators of branch point twist fields, here denoted by and . These are symmetry fields associated to cyclic permutation symmetry in a replica theory and having the smallest conformal dimension at the critical point. Recently, other twist fields (composite twist fields), typically of higher dimension, have been shown to play a role in the study of a new measure of entanglement known as the symmetry resolved entanglement entropy. In this paper we give an exact expression for the two-point function of a composite twist field that arises in the Ising field theory. In doing so we extend the techniques originally developed for the standard branch point twist field in free theories as well as an existing computation due to Horváth and Calabrese of the same two-point function which focused on the leading large-distance contribution. We study the ground state two-point function of the composite twist field and its conjugate . At criticality, this field can be defined as the leading field in the operator product expansion of and the disorder field µ. We find a general formula for and for (the derivative of) its analytic continuation to positive real replica numbers greater than 1. We check our formula for consistency by showing that at short distances it exactly reproduces the expected conformal dimension.

Symmetry-resolved entanglement of two-dimensional symmetry-protected topological states

Entanglement spectra and their symmetries provide unique fingerprints of symmetry-protected topological states (SPTs). Unfortunately, using such calculations to identify SPTs numerically is notoriously difficult. Here, the authors develop a numerical method mapping this computation to a one-dimensional problem by utilizing the local nature of the quantum circuits that generate any SPT. Its application to models with Ising symmetry fully confirms field-theoretic predictions for their entanglement spectrum. Additionally, nontrivial deformations of the SPT manifest as many-body Aharonov-Bohm fluxes in the entanglement spectrum.

Chaos and bipartite entanglement between Bose-Josephson junctions.

The entanglement between two weakly coupled bosonic Josephson junctions is studied in relation to the classical mixed phasespace structure of the system, containing symmetry-related regular islands separated by chaos. The symmetry-resolved entanglement spectrum and bipartite entanglement entropy of the system's energy eigenstates are calculated and compared to their expected structure for random states that exhibit complete or partial ergodicity. The entanglement spectra of chaos-supported eigenstates match the microcanonical structure of a Generalized Gibbs Ensemble due to the existence of an adiabatic invariant that restricts ergodization on the energy shell. The symmetry-resolved entanglement entropy of these quasistochastic states consists of a mean-field maximum entanglement term and a fluctuation correction due to the finite size of the constituent subsystems. The total bipartite entanglement entropy of the eigenstates correlates with their chaoticity. Island-supported eigenstates are macroscopic Schrödinger cat states for particles and excitations with substantially lower entanglement.

Symmetry resolved entanglement of excited states in quantum field theory. Part I. Free theories, twist fields and qubits

The excess entanglement resulting from exciting a finite number of quasiparticles above the ground state of a free integrable quantum field theory has been investigated quite extensively in the literature. It has been found that it takes a very simple form, depending only on the number of excitations and their statistics. There is now mounting evidence that such formulae also apply to interacting and even higher-dimensional quantum theories. In this paper we study the entanglement content of such zero-density excited states focusing on the symmetry resolved entanglement, that is on 1+1D quantum field theories that possess an internal symmetry. The ratio of charged moments between the excited and grounds states, from which the symmetry resolved entanglement entropy can be obtained, takes a very simple and universal form, which in addition to the number and statistics of the excitations, now depends also on the symmetry charge. Using form factor techniques, we obtain both the ratio of moments and the symmetry resolved entanglement entropies in complex free theories which possess U(1) symmetry. The same formulae are found for simple qubit states.